Consider the function f(x)=6x-cos(x)+5 on the interval 0 is less than or equal to x, and x is less than or equal to 1. The Intermediate Value Theorem guarantees that there is a value c such that for which values of c and k? Fill in the following mathematical statements, giving an interval in non-zero length in each case.

Question

Consider the function f(x)=6x-cos(x)+5 on the interval 0 is less than or equal to x, and x is less than or equal to 1. The Intermediate Value Theorem guarantees that there is a value c such that for which values of c and k? Fill in the following mathematical statements, giving an interval in non-zero length in each case.

For every k in _______less than or equal to k is less than or equal to _________ there is a c in _________ is less than or equal to c which is less than or equal to________ such as .

Thank you so much! Hopefully someone can figure this out! sorry if the wording is confusing, there are no "less than or equal to" signs on my computer.

Answer 1

Yoiu need f(x) at the endpoints:

f(0) = 4
f(1) = 11-cos(1) = 0.54

For every k in 0.54 <= k <= 4 there is a c in 0 <= c <= 1 such as .

That is, since f(x) changes continuously from 4 to .54, it assume every intermediate value at some point along the interval.

Answer 2

For every k in the interval -1 less than or equal to k less than or equal to 12, there is a c in the interval 0 less than or equal to c less than or equal to 1 such that .

Answer 3

Yoiu need f(x) at the endpoints:

For every k in the interval -1 less than or equal to k less than or equal to 12, there is a c in the interval 0 less than or equal to c less than or equal to 1 such that .

For every k in the interval [-1, 12] there is a c in the interval [0, 1] such that .